A Helmholtz equation solver using unsupervised learning: Application to transcranial ultrasound
This addresses the need for real-time acoustic field predictions in non-invasive brain treatments, representing an incremental improvement over existing methods.
The paper tackled the computational expense of conventional numerical wave solvers for transcranial ultrasound therapy by developing a fast iterative solver for the heterogeneous Helmholtz equation in 2D using a fully-learned optimizer, achieving excellent performance and generalization to larger domains and complex distributions like CT images.
Transcranial ultrasound therapy is increasingly used for the non-invasive treatment of brain disorders. However, conventional numerical wave solvers are currently too computationally expensive to be used online during treatments to predict the acoustic field passing through the skull (e.g., to account for subject-specific dose and targeting variations). As a step towards real-time predictions, in the current work, a fast iterative solver for the heterogeneous Helmholtz equation in 2D is developed using a fully-learned optimizer. The lightweight network architecture is based on a modified UNet that includes a learned hidden state. The network is trained using a physics-based loss function and a set of idealized sound speed distributions with fully unsupervised training (no knowledge of the true solution is required). The learned optimizer shows excellent performance on the test set, and is capable of generalization well outside the training examples, including to much larger computational domains, and more complex source and sound speed distributions, for example, those derived from x-ray computed tomography images of the skull.